By Sudhir R. Ghorpade, Balmohan V. Limaye

This ebook presents a self-contained and rigorous advent to calculus of capabilities of 1 variable. The presentation and sequencing of subject matters emphasizes the structural improvement of calculus. while, due value is given to computational concepts and purposes. The authors have strived to make a contrast among the intrinsic definition of a geometrical suggestion and its analytic characterization. in the course of the e-book, the authors spotlight the truth that calculus presents a company origin to numerous thoughts and effects which are often encountered in highschool and accredited on religion. for instance, you will discover right here an explanation of the classical end result that the ratio of the circumference of a circle to its diameter is similar for all circles. additionally, this ebook is helping scholars get a transparent figuring out of the idea that of an perspective and the definitions of the logarithmic, exponential and trigonometric services including an evidence of the truth that those will not be algebraic capabilities. a few themes which can were inadequately lined in calculus classes and glossed over in genuine research classes are handled the following in huge aspect. As such, this booklet offers a unified exposition of calculus and actual analysis.

The in basic terms necessities for analyzing this publication are themes which are often coated in highschool; besides the fact that, the reader is predicted to own a few mathematical adulthood and a capability to appreciate and have fun with proofs. This booklet can be utilized as a textbook for a significant undergraduate path in calculus, whereas elements of the e-book can be utilized for complicated undergraduate and graduate classes in actual research. each one bankruptcy comprises numerous examples and a wide collection of routines, in addition to "Notes and Comments" describing salient positive factors of the exposition, similar advancements and references to proper literature.

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12. 17. Show that if n ∈ N and a1 , . . , an are nonnegative real numbers, then (a1 +· · ·+an )2 ≤ n(a21 +· · ·+a2n ). ] 18. Let n and m be nonnegative integers. Show that there is an injective map from {1, . . , n} to {1, . . , m} if and only if n ≤ m. Also show that there is a surjective map from {1, . . , n} to {1, . . , m} if and only if n ≥ m. Deduce that there is a bijective map from {1, . . , n} to {1, . . , m} if and only if n = m. ) 19. Given any function f : R → R, prove the following: (i) If f (xy) = f (x) + f (y) for all x, y ∈ R, then f (x) = 0 for all x ∈ R.

Let p(x), q(x) ∈ R[x] be nonzero polynomials of degrees m and n respectively. Let e ∈ N be such that m < en. Show that there are polynomials A1 (x), . . , Ae (x) in R[x] such that either Ai (x) = 0 or deg Ai (x) < n for i = 1, . . , e and p(x) = A1 (x)q(x)e−1 + A2 (x)q(x)e−2 + · · · + Ae (x). Deduce that p(x) Ae (x) A1 (x) A2 (x) + ···+ . = + 2 q(x) q(x) q(x) q(x)e (Hint: Let R0 (x) := p(x). Use Exercise 58 to find A1 (x), . . , Ae (x) and R1 (x), . . , Re (x) successively in such a way that Ri−1 (x) = Ai (x)q(x)e−i + Ri (x) for i = 1, .

We would soon land in a vicious circle! Mathematicians find a way out of this dilemma by agreeing to regard certain terms as undefined or primitive. Further, one also stipulates certain axioms or postulates that describe some ‘natural’ properties that the primitive terms possess. Once this is done, every other term is defined using the primitive terms or the ones defined earlier. Also, a result is not accepted unless it is precisely proved using the axioms or the results proved before. The terms that are usually considered primitive or undefined in mathematics are “set” and “is an element of ” (a set).